Question

Let $r\left(t\right)=\left(x\left(t\right),y\left(t\right)\right),t\in [a,\infty ),$ be a vector-valued function, where a is a real number. Explain why the graph of r may or may not be contained in a circle centered at the origin. (Hint: Graph the functions $r_1\left(t\right)=\left(\frac{1}{t},\frac{1}{t}\right)$and $r_2\left(t\right)=\left(t,t\right),$both with domain [1,∞).)

Short answer

The graph of r may be contained in a circle centered at the origin.

Step-by-step solution

Step 1. Given Information.

The given vector-valued function is$r\left(t\right)=\left(x\left(t\right),y\left(t\right)\right),t\in [a,\infty ).$

Step 2. Explanation.

To explain the graph of r may or may not be contained in a circle centered at the origin, let $r_1\left(t\right)=\left(\frac{1}{t},\frac{1}{t}\right)\mathrm{with}\mathrm{domain}[1,\infty ).$

Now, the graph of $r_1\left(t\right)$is

Step 3. Explanation.

Now, let $r_2\left(t\right)=\left(t,t\right)\mathrm{with}\mathrm{domain}[1,\infty ).$

The graph of $r_2\left(t\right)$is

From both graphs, we can depictr that may be contained in a circle centered at the origin.